12010, May 18th
Tokyo Tech.a Univ. of Tokyo/RIKENb TU Münchenc
KN interaction and Λ*N interaction in chiral SU(3) dynamics
Tetsuo Hyodoa,Yoichi Ikedab, Wolfram Weisec,
Toshitaka Uchinoa, and Makoto Okaa
supported by Global Center of Excellence Program“Nanoscience and Quantum Physics”
2
Contents
Introduction to chiral SU(3) dynamics
KN interaction in chiral SU(3) dynamics
Λ*N interaction in chiral SU(3) dynamics
Summary
Contents
・Λ*N potential in meson-exchange picture・structure of the Λ*N bound state
・structure of the Λ* = Λ(1405)・toward the “realistic KN-πΣ interaction”
3
Chiral symmetry breaking in hadron physics
Consequence of chiral symmetry breaking in hadron physics
Chiral symmetry and its breaking
Underlying QCD <==> observed hadron phenomenaSU(3)R ⊗ SU(3)L → SU(3)V
Introduction to chiral SU(3) dynamics
Chiral symmetry: QCD with massless quarks
- constraints on the interaction of NG boson and a hadron low energy theorems <-- current algebra systematic low energy (m,p/4πfπ) expansion: ChPT
- appearance of the Nambu-Goldstone (NG) boson
- dynamical generation of hadron massesmπ ∼ 140 MeV
Mp ∼ 1 GeV ∼ 3Mq, Mq ∼ 300 MeV v.s. 3− 7 MeV
4
s-wave low energy interactionLow energy NG boson (Ad)- target hadron (T) scattering
Projection onto s-wave: Weinberg-Tomozawa termY. Tomozawa, Nuovo Cim. 46A, 707 (1966); S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)
Cα,T = �2FT · FAd� = C2(T ) − C2(α) + 3
Group theoretical structure and flavor SU(3) symmetry determines the sign and the strength of the interaction
Cij =�
α
Cα,T
�8 T α
IMi , YMi ITi , YTi I, Y
��8 T α
IMj , YMj ITj , YTj I, Y
�
Low energy theorem: leading order term in ChPT
Introduction to chiral SU(3) dynamics
Vij = −Cij
4f2(ωi + ωj) energy dependence (derivative coupling)
decay constant of π (gV=1)
5
Scattering amplitude and unitarityUnitarity of S-matrix: Optical theorem
Im [T−1(s)] =ρ(s)2
phase space of two-body state
General amplitude by dispersion relation
T−1(√
s) =�
i
Ri√s−Wi
+ a(s0) +s− s0
2π
� ∞
s+ds�
ρ(s�)(s� − s)(s� − s0)
Scattering amplitude
V? chiral expansion of T, (conceptual) matching with ChPTT (1) = V (1), T (2) = V (2), T (3) = V (3) − V (1)GV (1), ...
Amplitude T: consistent with chiral symmetry + unitarity
Ri, Wi, a: to be determined by chiral interaction
Identify dispersion integral = loop function G, the rest = V-1
T (√
s) =1
V −1(√
s)−G(√
s; a)
Introduction to chiral SU(3) dynamics
6
Description of S = -1, KN s-wave scattering: Λ(1405) in I=0Chiral unitary approach
R.H. Dalitz, T.C. Wong, G. Rajasekaran, Phys. Rev. 153, 1617 (1967)
Y. Tomozawa, Nuovo Cim. 46A, 707 (1966); S. Weinberg, Phys. Rev. Lett. 17, 616 (1966)- Interaction <-- chiral symmetry
- Amplitude <-- unitarity in coupled channels
Introduction to chiral SU(3) dynamics
T
= +T
N. Kaiser, P. B. Siegel, W. Weise, Nucl. Phys. A594, 325 (1995),E. Oset, A. Ramos, Nucl. Phys. A635, 99 (1998),J. A. Oller, U. G. Meissner, Phys. Lett. B500, 263 (2001),M.F.M. Lutz, E. E. Kolomeitsev, Nucl. Phys. A700, 193 (2002), .... many others
It works successfully, also in S=0 sector, meson-meson scattering sectors, systems including heavy quarks, ...
chiral cutoffT =
11− V G
V
7
T. Hyodo, S.I. Nam, D. Jido, A. Hosaka, PRC68, 018201 (2003); PTP 112, 73 (2004)
200
150
100
50
0
!T [
mb]
300200100
K-p70
60
50
40
30
20
10
0300200100
"0#
200
150
100
50
0300200100
"+$%
60
50
40
30
20
10
0
!T [
mb]
300200100Plab [MeV/c]
K0n60
50
40
30
20
10
0300200100
Plab [MeV/c]
"0$080
60
40
20
0300200100
Plab [MeV/c]
"%$+
The simplest model (1 parameter) v.s. experimental data
14401420140013801360s [MeV]
mas
s dist
ribut
ion
γ Rc Rn
exp. 2.36 0.664 0.189
theo. 1.80 0.624 0.225
Total cross section of K-p scattering Branching ratio
πΣ spectrum
Λ(1405)
Good agreement with data above, at, and below KN thresholdΛ(1405) mass, width, couplings: prediction of the model
KN interaction in chiral SU(3) dynamics
8
Two poles for one resonancePoles of the amplitude in the complex plane: resonance
∼
Tij(√
s) ∼ gigj√s − MR + iΓR/2
13601380
14001420
1440
-20-40
-60-80
0.5
1.0
1.5
0.5
1.0
1.5
!(1405)
Re[z] Im[z]
|T|
D. Jido, J.A. Oller, E. Oset, A. Ramos, U.G. Meissner, Nucl. Phys. A 723, 205 (2003);T. Hyodo, W. Weise, Phys. Rev. C 77, 035204 (2008)
short summary- Λ(1405): Λ*1, Λ*2- Λ*i masses, widths, Λ*i-MB couplings predicted
KN interaction in chiral SU(3) dynamics
Physical Λ* include two poles KN bound state + πΣ resonance
Λ*1Λ*2
c.f.) Gengʼs talk
Very strong attraction in KN (higher energy) --> bound stateStrong attraction in πΣ (lower energy) --> resonance
-120
-100
-80
-60
-40
-20
0
Im z
[MeV
]
1440140013601320Re z [MeV]
resonance
KN bound stateWithout off-diagonal
KNπΣ
9
Origin of the two-pole structure
KN πΣ
Leading order chiral interaction
-120
-100
-80
-60
-40
-20
0
Im z
[MeV
]
1440140013601320Re z [MeV]
resonance
z2(full)
z1(full)
KN bound state
KNπΣ
T. Hyodo, W. Weise, Phys. Rev. C 77, 035204 (2008)
KN interaction in chiral SU(3) dynamics
Vij = −Cijωi + ωj
4f2
Cij =
3 −�
32
−�
32 4
ωi ∼ mi, 3.3mπ ∼ mK
⇒ VKN ∼ 2.5VπΣ
at threshold
10
“realistic KN-πΣ interaction”Position of the second pole: model dependent- higher order terms in the interaction kernel?- energy dependence? (c.f. Ikedaʼs talk)
--> uncertainties in KN interaction below threshold
KN interaction in chiral SU(3) dynamics
KNKN scatt.?
πΣ energy
Systematic χ2 study including NLO interactionsY. Ikeda, T. Hyodo, W. Weise, in preparation
- Precise data of KN scattering length <-- SIDDHARTA- Precise data of πΣ spectrum <-- FOPI (this afternoon?)- Any information of πΣ scattering length <-- lattice QCD?
==> the “realistic KN-πΣ interaction”
11
K in nuclei- KN interaction is strongly attractive <-- Λ(1405). Formation of (deeply) bound kaonic nuclei?
- Structure of the Λ(1405), kaon condensation, ...
The simplest K-nucleus: KNN three-body system
Y. Akaishi, T. Yamazaki, Phys. Rev. C65, 044005 (2002)
Theory: rigorous few-body calculations with realistic interactions
- System bounds - Quantitative difference: uncertainties in KN int. at far below threshold
Yamazaki-Akaishi, Shevchenko, et al., Ikeda-Sato, Doté et al., Wycech-Green,
Experiment: some “evidences” in ΛN mass spectra
- Interpretation (πΣN?FSI?)
FINUDA, DISTO, OBELIX, etc.
FINUDA05
Λ*N interaction in chiral SU(3) dynamics
12
Λ* hypernuclei model
A. Arai, M. Oka, S. Yasui, Prog. Theor. Phys. 119, 103 (2008)
Regarding a KN(I=0) pair as the Λ*=Λ(1405), construct the “Λ*N potential” with the meson-exchange picture
To determine the coupling and make predictions, we need a framework to describe the Λ* <-- chiral unitary approach
Λ* coupling constants: unknown (<-- FINUDA data).
T. Yamazaki, Y. Akaishi, Phys. Rev. C76, 045201 (2007),A. Doté, T. Hyodo, W. Weise, Nucl. Phys. A 804, 197 (2008); Phys. Rev. C 79, 014003 (2009)
<== Λ* structure seems to be surviving in KNN system.
Λ*N interaction in chiral SU(3) dynamics
13
Λ*N potential with meson-exchange pictureΛ*iN potential with one boson exchange (i=1,2)
NNσ, NNω couplings: Jülich (model A) YN potential Λ*iKN coupling: chiral unitary approach (pole residue) Λ*iΛ*iσ, Λ*iΛ*iω, couplings --> estimated by microscopic MB=(KN,πΣ) couplings
+
σ decay to ππgKKσ = 0
Λ*N interaction in chiral SU(3) dynamics
Binding
Binding
No mixing With mixing
Mixing
Mixing
14
Λ*N potential: mixing interactionChiral unitary approach --> two Λ* states : Λ*1, Λ*2 With sufficient attraction, two Λ*N bound states in B=2 system : Λ*1N, Λ*2N
There can be the mixing of Λ*1N <--> Λ*2N
Binding
Binding
No mixing With mixing
Mixing
Mixing
Λ*N interaction in chiral SU(3) dynamics
15
Λ*N potential: each contributionDiagonal potentials for Λ*2N in S=0 and S=1, s-wave
S=0: attractive pocket at intermediate range <-- K exchange is attractiveS=1: no intermediate attraction, but a short range dip <-- K exchange is repulsive
-400
-200
0
200
400
V(r)
[MeV
]
2.01.51.00.50.0
r [fm]
Total K
S = 1, (1426)
-400
-200
0
200
400
V(r)
[MeV
]
2.01.51.00.50.0
r [fm]
Total K
S = 0, (1426)
Qualitatively similar potential for Λ*1N
Λ*N interaction in chiral SU(3) dynamics
16
Λ*N bound states without mixingSolve the schrödinger equation for the s-wave Λ*iN potential without the mixing interaction.
- no physical bound states in S=1 channels (consistent with the analysis of the volume integral)
binding from
Λ*N th. [MeV]
KNN th. [MeV]
Λ*2 N 13.39 22.39
Λ*1 N 0.34 45.34
- for S=0 we obtain the bound states in both Λ*i
No mixing
0.3 MeV
45 MeV
22 MeV13 MeV
A. Doté, T. Hyodo, W. Weise, Nucl. Phys. A 804, 197 (2008); Phys. Rev. C 79, 014003 (2009)Similar value (20±3 MeV) in KNN single-channel approach
Λ*N interaction in chiral SU(3) dynamics
17
Λ*N bound states with mixingWith mixing, the higher state becomes a resonance.Real scaling method ≈ changing the box sizeλ: strength of the mixing interaction, physical for λ=1
The lower energy state bounds more.The higher energy state disappears (above Λ*2N threshold?).
λ=0-20
-10
10
20
30
40
50
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Binding Energy
-20
-10
10
20
30
40
50
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Binding Energy
λ=1/4-20
-10
10
20
30
40
50
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Binding Energy
λ=1/2-20
-10
10
20
30
40
50
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Binding Energy
λ=3/4-20
-10
10
20
30
40
50
0.8 1 1.2 1.4 1.6 1.8 2 2.2
Binding Energy
λ=1
resonance (compact object)--> stable against the box size
Λ*N interaction in chiral SU(3) dynamics
α
18
Result of Λ*N bound states
taken from T. Uchino, Master thesis
- The Λ*N bound state decays into πΣN and YN.- Disappeared state could be a very broad state.- Physical Λ* (especially the lower one) has a finite width.- Comparison with three-body calculation (Ikda et al)
Result in one figure:
(Λ∗N) ∼ 0.7(Λ∗1N) + 0.3(Λ∗
2N)
∼ 0.5(KNN) + 0.5(πΣN)
substantial mixture of πΣN component
decomposition of constituents
Λ*N interaction in chiral SU(3) dynamics
19
Summary: KN interaction
Chiral low energy theorem: constrains for the NG boson dynamicsTwo poles for the Λ(1405)
Systematic study for the KN-πΣ system
We study the KN-πΣ system and Λ(1405) based on chiral SU(3) symmetry and unitarity
Summary
Y. Ikeda, T. Hyodo, W. Weise, in preparation
<-- new data, such as KN scattering length, πΣ spectrum are called for.
<-- attractive KN and πΣ interactionsT. Hyodo and W. Weise, Phys. Rev. C 77, 035204 (2008)
20
Summary: Λ*N interaction
Chiral dynamics: two states Λ*1, Λ*2
Both Λ*i generate bound states with N in spin S=0 channels <-- K exchange
With the mixing, the lower state bounds more, and the higher state dissolves.
We study the Λ*N two-body system based on the Λ*N potential with chiral SU(3) dynamics.
Summary
mass of Λ*N ~ 2316-2322 MeV<-- substantial mixture of πΣN
T. Uchino, T. Hyodo, M. Oka, in preparation